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In the figure,   AD = CD and AB = CB . 

(i) Prove that Line BD bisects angle  ADC (i.e. that line BD cuts angle  ADC into two equal angles).

(ii) Prove that Line AC and Line DB are perpendicular.

 

Maplesnowy  Mar 13, 2018
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Since  AD  = CD

Then the angles opposite these sides are also equal

So angle DAC  = angle DCA

 

And if AB = CB, then the angles opposite thes sides are also equal

So angle CAB  = angle ACB

 

So angle DAC + angle CAB  = angle DCA + angle ACB

So angle DAB   = angle CDB

 

Then, by  SAS....triangle  DAB  = triangle DCB

 

Then angle ADB  = angle CDB.....so angle ADC is bisected

 

And angle  DAC  = angle DCA

AD = CD

And angle DAC  = angle DCA

So....by ASA....triangle EAD  = triangle   ECD

 

So angle AED  = angle CED

 

And, by Euclid.....a line standing upon a line that makes adjacent angles equal means that the lines are perpendicular....so AC  and Db are perpendicular

 

 

cool cool cool

CPhill  Mar 13, 2018

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